Loci of Point(Thedirectdata.com)

8/22/2019 Loci of Point(Thedirectdata.com)

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LOCUSIt is a path traced out by a point moving in a plane,

in a particular manner, for one cycle of operation.

The cases are classified in THREE categories for easy understanding.

A} Basic Locus Cases.

B} Oscillating Link

C} Rotating Link

Basic Locus Cases:Here some geometrical objects like point, line, circle will be described with there relative

Positions. Then one point will be allowed to move in a plane maintaining specific relation

with above objects. And studying situation carefully you will be asked to draw its locus.Oscillating & Rotating Link:Here a link oscillating from one end or rotating around its center will be described.

Then a point will be allowed to slide along the link in specific manner. And now studying

the situation carefully you will be asked to draw its locus.

STUDY TEN CASES GIVEN ON NEXT PAGES


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A

B

p

4 3 2 1F

1 2 3 4

SOLUTION STEPS:

1.Locate center of line, perpendicular to

AB from point F. This will be initial

point P.

2.Mark 5 mm distance to its right side,

name those points 1,2,3,4 and from those

draw lines parallel to AB.

3.Mark 5 mm distance to its left of P andname it 1.

4.Take F-1 distance as radius and F as

center draw an arc

cutting first parallel line to AB. Name

upper point P1 and lower point P2.

5.Similarly repeat this process by taking

again 5mm to right and left and locateP3P4.

6.Join all these points in smooth curve.

It will be the locus of P equidistance

from line AB and fixed point F.

P1

P2

P3

P4

P5

P6

P7

P8

PROBLEM 1.: Point F is 50 mm from a vertical straight line AB.

Draw locus of point P, moving in a plane such that

it always remains equidistant from point F and line AB.

Basic Locus Cases:


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A

B

p

4 3 2 1 1 2 3 4

P1

P2

P3

P4

P5

P6

P7

P8

C

SOLUTION STEPS:

1.Locate center of line, perpendicular to

AB from the periphery of circle. This

will be initial point P.

2.Mark 5 mm distance to its right side,

name those points 1,2,3,4 and from those

draw lines parallel to AB.

3.Mark 5 mm distance to its left of P andname it 1,2,3,4.

4.Take C-1 distance as radius and C as

center draw an arc cutting first parallel

line to AB. Name upper point P1 and

lower point P2.

5.Similarly repeat this process by taking

again 5mm to right and left and locateP3P4.

6.Join all these points in smooth curve.

It will be the locus of P equidistance

from line AB and given circle.

50 D

75 mm

PROBLEM 2 :

A circle of 50 mm diameter has its center 75 mm from a vertical

line AB.. Draw locus of point P, moving in a plane such that

it always remains equidistant from given circle and line AB.

Basic Locus Cases:


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95 mm

30 D

60 D

p4 3 2 1 1 2 3 4

C2C1

P1

P2

P3

P4

P5

P6

P7

P8

PROBLEM 3 :

Center of a circle of 30 mm diameter is 90 mm away from center of another circle of 60 mm diameter.

Draw locus of point P, moving in a plane such that it always remains equidistant from given two circles.

SOLUTION STEPS:

1.Locate center of line,joining twocenters but part in between periphery

of two circles.Name it P. This will be

initial point P.

2.Mark 5 mm distance to its right

side, name those points 1,2,3,4 and

from those draw arcs from C1As center.

3. Mark 5 mm distance to its rightside, name those points 1,2,3,4 and

from those draw arcs from C2 As

center.

4.Mark various positions of P as per

previous problems and name those

similarly.

5.Join all these points in smooth

curve.

It will be the locus of P

equidistance from given two circles.

Basic Locus Cases:


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2CC1

30 D

60 D

350

C1

Solution Steps:1) Here consider two pairs,

one is a case of two circles

with centres C1 and C2 and

draw locus of point P

equidistance from

them.(As per solution of

case D above).

2) Consider second casethat of fixed circle (C1)

and fixed line AB and

draw locus of point P

equidistance from them.

(as per solution of case B

above).

3) Locate the point where

these two loci intersect

each other. Name it x. It

will be the point

equidistance from given

two circles and line AB.

4) Take x as centre and its

perpendicular distance on

AB as radius, draw a circle

which will touch given two

circles and line AB.

Problem 4:In the given situation there are two circles of

different diameters and one inclined line AB, as shown.

Draw one circle touching these three objects.

Basic Locus Cases:


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PA B

4 3 2 1 1 2 3 4

70 mm 30 mm

p1

p2

p3

p4

p5

p6

p7

p8

Problem 5:-Two points A and B are 100 mm apart.

There is a point P, moving in a plane such that the

difference of its distances from A and B always

remains constant and equals to 40 mm.

Draw locus of point P.

Basic Locus Cases:

Solution Steps:1.Locate A & B points 100 mm apart.

2.Locate point P on AB line,

70 mm from A and 30 mm from B

As PA-PB=40 ( AB = 100 mm )

3.On both sides of P mark points 5

mm apart. Name those 1,2,3,4 as usual.

4.Now similar to steps of Problem 2,

Draw different arcs taking A & B centers

and A-1, B-1, A-2, B-2 etc as radius.

5. Mark various positions of p i.e. and join

them in smooth possible curve.

It will be locus of P


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1) Mark lower most

position of M on extension

of AB (downward) by taking

distance MN (40 mm) from

point B (because N cannot go beyond B ).

2) Divide line (M initial

and M lower most ) into

eight to ten parts and mark

them M1, M2, M3 up to the

last position of M .

3) Now take MN (40 mm)

as fixed distance in compass,

M1 center cut line CB in N1.4) Mark point P1 on M1N1

with same distance of MP

from M1.

5) Similarly locate M2P2,

M3P3, M4P4 and join all P

points.

It will be locus of P.

Solution Steps:

600

M

N

N1

N2

N3

N4

N5N6

N7N8

N9

N10

N11

N12

A

B

C

D

M1

M2

M3

M4

M5

M7

M8

M9

M10

M11

M6

M12

M13

N13

p

p1

p2

p3

p4p5

p6

p7

p8

p9

p10

p13

p11

p12

Problem 6:-Two points A and B are 100 mm apart.

There is a point P, moving in a plane such that the

difference of its distances from A and B always

remains constant and equals to 40 mm.


FORK & SLIDER


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1

2

3

4

5

6

7

8

p

p1

p2p3

p4

p5

p6

p7

p8

O

A A1

A2

A3

A4

A5

A6

A7

A8

Problem No.7:

A LinkOA, 80 mm long oscillates around O,

600to right side and returns to its initial vertical

Position with uniform velocity.Mean while point

P initially on O starts sliding downwards and

reaches end A with uniform velocity.

Draw locus of point P

Solution Steps:Point P- Reaches End A (Downwards)

1) Divide OA in EIGHT equal parts and from O to A after O

name 1, 2, 3, 4 up to 8. (i.e. up to point A).

2) Divide 600 angle into four parts (150 each) and mark each

point by A1, A2, A3, A4 and for return A5, A6, A7 andA8.

(Initial A point).3) Take center O, distance in compass O-1 draw an arc upto

OA1. Name this point as P1.

1) Similarly O center O-2 distance mark P2 on line O-A2.

2) This way locate P3, P4, P5, P6, P7 and P8 and join them.

( It will be thw desired locus of P )

OSCILLATING LINK


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p

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16O

A

Problem No 8:

A LinkOA, 80 mm long oscillates around O,

600

to right side, 1200

to left and returns to its initialvertical Position with uniform velocity.Mean while point

P initially on O starts sliding downwards, reaches end A

and returns to O again with uniform velocity.

Draw locus of point P

Solution Steps:( P reaches A i.e. moving downwards.

& returns to O again i.e.moves upwards )

1.Here distance traveled by point P is PA.plus

AP.Hence divide it into eight equal parts.( so

total linear displacement gets divided in 16

parts) Name those as shown.

2.Link OA goes 600 to right, comes back to

original (Vertical) position, goes 600 to left

and returns to original vertical position. Hencetotal angular displacement is 2400.

Divide this also in 16 parts. (150 each.)

Name as per previous problem.(A, A1 A2 etc)

3.Mark different positions of P as per the

procedure adopted in previous case.

and complete the problem.

A2

A1

A3

A4

A5

A6

A7A8

A9

A10

A11

A12

A13

A14

A15

A16

p8

p5

p6

p7

p2p4

p1p

3

OSCILLATING LINK


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A B

A1

A2

A4

A5

A3

A6

A7

P

p1 p2

p3

p4

p5

p6 p7

p8

1 2 34 5 6 7

Problem 9:

Rod AB, 100 mm long, revolves in clockwise direction for one revolution.

Meanwhile point P, initially on A starts moving towards B and reaches B.


ROTATING LINK

1) AB Rod revolves around

center O for one revolution andpoint P slides along AB rod andreaches end B in onerevolution.2) Divide circle in 8 number ofequal parts and name in arrowdirection after A-A1, A2, A3, upto A8.3) Distance traveled by point P

is AB mm. Divide this also into 8number of equal parts.4) Initially P is on end A. When

A moves to A1, point P goesone linear division (part) awayfrom A1. Mark it from A1 andname the point P1.5) When A moves to A2, P willbe two parts away from A2(Name it P2 ). Mark it as abovefrom A2.6) From A3 mark P3 threeparts away from P3.7) Similarly locate P4, P5, P6,P7 and P8 which will be eightparts away from A8. [Means Phas reached B].

8) Join all P points by smoothcurve. It will be locus of P


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A B

A1

A2

A4

A5

A3

A6

A7

P

p1

p2

p3

p4

p5

p6

p7

p81 2 3 4567

Problem 10 :

Rod AB, 100 mm long, revolves in clockwise direction for one revolution.

Meanwhile point P, initially on A starts moving towards B, reaches B

And returns to A in one revolution of rod.


Soluti on Steps

+ + + +

ROTATINGLINK

1) AB Rod revolves around center Ofor one revolution and point P slidesalong rod AB reaches end B andreturns to A.2) Divide circle in 8 number of equalparts and name in arrow directionafter A-A1, A2, A3, up to A8.3) Distance traveled by point P is ABplus AB mm. Divide AB in 4 parts sothose will be 8 equal parts on return.4) Initially P is on end A. When Amoves to A1, point P goes onelinear division (part) away from A1.Mark it from A1 and name the pointP1.5) When A moves to A2, P will be

two parts away from A2 (Name it P2). Mark it as above from A2.6) From A3 mark P3 three partsaway from P3.7) Similarly locate P4, P5, P6, P7and P8 which will be eight parts awayfrom A8. [Means P has reached B].8) Join all P points by smooth curve.It will be locus of P

The Locus will follow the looppath two times in one revolution.

Loci of Point(Thedirectdata.com)

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